A field guide  ·  three interactive monographs

Time Series, end to end

One arc told in three parts: from a single sensor sample, to a calibrated forecast of its next value, to a forecast of the failure it is heading toward. Every idea is built from first principles and made tangible with genuine, interactive simulations you can drive yourself.

By Majid Mazouchi

Three parts: sensing the signal, forecasting its next value, and forecasting its failure. part one · sensingsignal · spectrum · residual part two · forecastingbaselines · intervals · ML part three · prognosisdegradation · RUL · risk the same signal, read at three horizons: the present, the next step, and the end of life

The three parts

Start anywhere, but they build

Each monograph stands on its own, yet they share a vocabulary and a spine. Read in order for the full arc, or jump to the part you need.

Part one

Time Series as Sensory Data

How a physical signal becomes ordered numbers, and how to read them: sampling and aliasing, decomposition, the frequency view, autocorrelation, smoothing and envelope analysis, feature extraction and fusion, the Kalman filter and its innovations, alarms, and a first look at degradation. The raw materials for everything that follows.

sampling · aliasing · FFT · spectrogram · autocorrelation · envelope · Kalman · CUSUM · features

read part one → Part two

Forecasting Time Series

From the past to the next value. The baselines you must beat, honest backtesting and scale-free metrics, exponential smoothing and ARIMA, reading the ACF and PACF, differencing toward stationarity, decomposition forecasts, and a deep run at uncertainty: fan charts, conformal intervals, quantile forecasting, and calibration. Then machine learning, direct versus recursive, forecast combination, deep and foundation models, and a live leaderboard.

baselines · MASE · ARIMA · ACF/PACF · conformal · quantile · calibration · ensembles · leaderboard

read part two → Part three

Prognostics and Remaining Useful Life

How long until it fails, and how sure are you. Building a health indicator, fitting and extrapolating degradation models, tracking the state with a Kalman filter, particle filters for nonlinear wear, similarity-based RUL from a failure library, honest conformal RUL intervals with the alpha-lambda funnel, operating conditions as covariates, and how to grade a prognostic when being late is far worse than being early.

health indicator · degradation · particle filter · similarity · conformal RUL · alpha-lambda · covariates

read part three →

Shared glossary

The vocabulary of the series

Terms that recur across the three parts, in plain language. The tag shows where each is introduced.

part one part two part three
Sampling & aliasingP1
Turning a continuous signal into samples at a fixed rate. Sample too slowly and high frequencies masquerade as low ones, an irreversible corruption called aliasing.
StationarityP2
A series whose mean, variance and correlations do not drift over time. Most classical forecasting assumes it, and differencing is the usual route to it.
Autocorrelation (ACF)P1
How strongly a series correlates with itself at each lag. The partial version, the PACF, isolates the direct link at a lag after removing the shorter ones in between.
DecompositionP1
Splitting a series into trend, seasonal and remainder parts, each easier to understand, model and forecast on its own.
Residual / innovationP1
What a model failed to predict: the gap between observation and expectation. A surprisingly large one is the raw material of both detection and prognosis.
Kalman filterP1
A recursive estimator that fuses a model prediction with each new measurement, tracking a hidden state and its uncertainty. Reused to project degradation forward in part three.
Particle filterP3
A sampling estimator that represents belief as a weighted cloud of state hypotheses, handling nonlinear and non-Gaussian problems the Kalman filter cannot.
BaselineP2
A trivially simple forecast such as repeating the last value. The bar every real method must clear; if it cannot, it has earned nothing.
BacktestingP2
Evaluating a forecaster only on data it never saw, by rolling the train/test split forward through history. The single most important discipline in the series.
MASEP2
A scale-free error metric that divides a method's error by the naive baseline's. Below one means you beat the baseline; above one means you did not.
Exponential smoothingP2
Forecasts built from weighted averages that fade into the past, with optional trend and seasonal terms. The Holt-Winters family.
ARIMAP2
A classical model combining autoregression, differencing and moving-average terms. Its orders are read from the ACF and PACF.
Prediction intervalP2
A range expected to contain the future value with a stated probability. The honest forecast is the interval, not just the central line.
Conformal predictionP2
A way to wrap any forecaster in an interval with a coverage guarantee that holds without trusting the model's assumptions, built from held-out residual quantiles. Central to RUL bounds in part three.
Quantile & pinball lossP2
Predicting percentiles directly rather than a mean, trained with the tilted pinball loss so intervals can be lopsided when reality is.
CalibrationP2
Whether intervals are honest: do ninety-percent intervals actually contain the truth ninety percent of the time. Checked with a reliability diagram.
Forecast combinationP2
Averaging several decent forecasts. Their errors partly cancel, so the ensemble is steadier than its parts, a famously hard baseline to beat.
Foundation modelP2
A network pretrained on a vast corpus of series that forecasts new ones zero-shot. Promising, but to be judged on the same backtest as anything else.
Health indicatorP3
A single number fused from features that climbs steadily as a machine wears, ideally monotonic, trendable and comparable across a fleet. The thing prognostics forecasts.
Remaining useful life (RUL)P3
The time left before a machine crosses its failure threshold. A forecast with a finish line, and the central quantity of part three.
Degradation modelP3
A parametric curve, linear, exponential or power-law, fitted to the health indicator and extrapolated to the threshold to estimate the RUL.
Similarity-based RULP3
Estimating remaining life by matching a partial degradation trajectory against a library of units that ran to failure, with no assumed curve.
Alpha-lambda accuracyP3
A prognostic standard: does the RUL estimate sit inside a band that narrows toward failure. The funnel a good predictor must enter and stay within.
Operating covariateP3
An external condition such as load or temperature that changes the degradation rate, so the same health state implies different lives under different futures.

A note on method

Why simulations, not screenshots

Every figure in this series is a live computation, not a recorded animation. The fan charts run real exponential smoothing, the particle filter genuinely resamples its cloud, the leaderboard actually backtests each method and sorts the results. When you move a slider, the mathematics re-runs in front of you. The aim is for the intuitions to be earned from a working model rather than asserted, because a method you have watched succeed and fail under your own hands is one you can trust and reach for later.

Each page is self-contained and runs offline in any modern browser. References on each page point to further study; the explanations and simulations are written and implemented from first principles.